# Mathematical Statements

__Mathematical Statement__

A meaningful composition of words which can be considered either true or false is called a mathematical statement or simply a statement.

A single letter shall be used to denote a statement. For example, the letter ‘** p**’ may be used to stand for the statement “

**is an equilateral triangle.” Thus,**

*ABC***is an equilateral triangle.**

*p = ABC*

__Truth Value of a Statement__

A statement is said to have truth value ** T **or

**according to whether the statement considered is true or false. For example, the statement ‘**

*F***2 plus 2 is four**’ has truth value

*T**,*whereas the statement ‘

**2 plus 2 is five**’ has truth value

*F**.*The knowledge of truth value of statements enables us to replace one statement by another (equivalent) statement(s).

__Production of New Statement__

New statements from given statements can be produced by:

**(i) Negation:** $$ \sim $$

If

**is a statement then its negation ‘$$ \sim $$**

*p***’ is statement ‘**

*p***not p**’. ‘$$ \sim $$

**’ has truth value**

*p***or**

*F***according to the truth value of ‘**

*T***’ is**

*p***or**

*T***.**

*F***(ii) Implication:** $$ \Rightarrow $$

If from a statement

**another statement**

*p***follows, we say ‘**

*q***p implies q**’ and write ‘

**’. Such a result is called an implication. The truth value of ‘**

*p*$$ \Rightarrow $$*q***’ is**

*p*$$ \Rightarrow $$*q***only when**

*F***has truth value**

*p***and**

*T***has the truth value**

*q***.**

*F*The statements involving ‘

**if p holds then q**’ are of the kind

**. For example, $$x = 2 \Rightarrow {x^2} = 4$$.**

*p*$$ \Rightarrow $$*q***(iii) Conjunction:** $$ \wedge $$

**The sentence ‘**

**p and q**’ which may be denoted by ‘

**’ is the conjunction of**

*p*$$ \wedge $$*q***and**

*p***. The truth value of**

*q***is T only when both**

*p*$$ \wedge $$*q***and**

*p***are true.**

*q***(iv) Disjunction:** $$ \vee $$

The sentence ‘

**p and q (or both)**’ which may be denoted by ‘

**’ is called the disjunction of the statements**

*p*$$ \vee $$*q***and**

*p***. The truth value of**

*q***is F only when both**

*p*$$ \vee $$*q***and**

*p***are false.**

*q*

__Equivalence of Two Statements__, *p*$$ \Leftrightarrow $$*q*

Two statements ** p** and

**are said to be**

*q***equivalent**if one implies the other, and in such a case we use the double implication symbol $$ \Leftrightarrow $$ and write

**.**

*p*$$ \Leftrightarrow $$*q*The statements which involve the phrase ‘**if and only if**’ or ‘is equivalent to’ or ‘**the necessary and sufficient conditions**‘ are of the kind ** p$$ \Leftrightarrow $$q**. For example,

**is an equilateral triangle**

*ABC***.**

*AB**= BC = CA*

For brevity, the phrase ‘**if and only if**’ is shortened to “**iff**”. As described above, the symbols $$ \vee $$ stand for the words ‘**and**’ and ‘**or**’ respectively. The disjunction symbol $$ \vee $$ is used in the logical sense ‘**and/or**’. The symbols $$ \wedge $$, $$ \vee $$ are logical connectives and are frequently used.

The following is the table showing truth values of different compositions of statements. Such tables are called truth tables.

p |
q |
$$ \sim $$
p |
$$ \sim $$
q |
p$$ \Rightarrow $$q |
p$$ \wedge $$q |
p$$ \vee $$q |
p$$ \Leftrightarrow $$q |

T |
T |
F |
F |
T |
T |
T |
T |

T |
F |
F |
T |
F |
F |
T |
F |

F |
T |
T |
F |
T |
F |
T |
F |

F |
F |
T |
T |
T |
F |
F |
T |

By forming truth tables, the equivalence of various statements can easily be ascertained. For example, we shall easily see that the implication ‘** p$$ \Rightarrow $$q**’ is equivalent to ‘$$ \sim $$

**’. The implication ‘$$ \sim $$**

*p*$$ \Rightarrow $$$$ \sim $$*q***’ is called the contrapositive of**

*q*$$ \Rightarrow $$$$ \sim $$*p***.**

*p*$$ \Rightarrow $$*q*